Index

∩, 9∪, 9

A close to Beigenvalues, 175

A invariant, 237Abel’s formula, 108, 252absolute convergence

convergence, 338adjugate, 85, 98algebraic number

minimal polynomial, 205algebraic numbers, 205

field, 206analytic function of matrix, 399Archimedean property, 20augmented matrix, 27

Banach space, 327basis, 63, 190Binet Cauchy

volumes, 292Binet Cauchy formula, 94block matrix, 104

multiplication, 105block multiplication, 104bounded linear transformations, 328

Cauchy interlacing theorem, 185, 187Cauchy Schwarz inequality, 33, 274, 325Cauchy sequence, 287, 327Cayley Hamilton theorem, 102, 251, 262centrifugal acceleration, 70centripetal acceleration, 70characteristic and minimal polynomial, 230characteristic equation, 143characteristic polynomial, 102, 228characteristic value, 143Cholesky factorization, 324codomain, 10cofactor, 83, 96column rank, 99, 116companion matrix, 254, 367complete, 347completeness axiom, 19complex conjugate, 14complex numbers

absolute value, 14field, 13

complex numbers, 13complex roots, 15composition of linear transformations, 224comutator, 185

condition number, 335conformable, 42conjugate linear, 279convex combination, 232convex hull, 231

compactness, 232coordinate axis, 31coordinates, 31Coriolis acceleration, 70Coriolis acceleration

earth, 72Coriolis force, 70counting zeros, 174Courant Fischer theorem, 303Cramer’s rule, 86, 87, 99cyclic basis, 242cyclic set, 239

defective, 148DeMoivre identity, 14dense, 21density of rationals, 21determinant

block upper triangular matrix, 161definition, 91estimate for Hermitian matrix, 272expansion along a column, 83expansion along a row, 83expansion along row, column, 96Hadamard inequality, 272inverse of matrix, 85matrix inverse, 97partial derivative, cofactor, 109permutation of rows, 92product, 94product of eigenvalues, 168, 178row, column operations, 85, 93summary of properties, 101symmetric definition, 93transpose, 93

diagonalizable, 222, 295minimal polynomial condition, 254basis of eigenvectors, 157

diagonalization, 300differentiable matrix, 66differential equations

first order systems, 181digraph, 44dimension of vector space, 192direct sum, 80, 234directed graph, 44discrete Fourier transform, 322

404